For instance, according to the supernatural wiki, silver is a metal with strong supernatural properties. But historically, why is specifically silver considered harmful to vampires? Is there a reason why it could not be gold or bronze?
First I'll get this out of the way: No research is truly comprehensive without TVTropes:
The myth of silver's mystical properties goes deep into human history. As a noble metal akin to gold, this is often attributed to something along the lines of silver's Incorruptible Pure Pureness. In fact, it's because of something people noticed a very long time ago: if you put water in a silver pitcher, it takes a lot longer for it to get unhealthily scummy. Silver has antimicrobial properties that make it quite useful in medicine. In ancient and medieval alchemy silver was also the metal with an affinity corresponding to the moon, so many of the mystical properties of the moon also became associated with the metal . . . Unlike in post-Medieval works, many pre-Christian pagan mythologies also associated it with the Sun, and many solar deities are described with silver objects (like Saule's silver thread, Apollo's silver bow and arrows or Amaterasu's silver mirror). Though ancient alchemists couldn't know it, silver does have a unique property among the elements — it exists in a naturally excited form — i.e. it has a single electron that exists in a higher energy level, leaving a gap in the shell immediately "below" it.
(I assume "unique among the elements" means rather "unique as an element that exists in amounts and malleability to the extent that humans can fashion discrete and widely-distributed objects with it" - however, I am not even an armchair chemist.)
As DukeZhou's comment indicates, The Strain develops a "hard sci-fi" angle to explain the connection in the case of a 'vampire species':
While silver poses low toxicity and risk to humans, certain bacteria and possibly fungi may be genetically susceptible to silver interfering with their enzyme production. A similar biological reaction to silver may be supernaturally present in strigoi. Regardless of the debated status on the medical effectiveness of silver, the retention of silver in the body associated with chronic intake of colloidal silver would prove disagreeable to vampires.
(If I could participate in Meta yet I would have checked to see how (un)acceptable it is to include a 'retcon' on this level, i.e. 21st century fiction employing/explaining folklore. However, since there were apparently still Romanians flouting a law banning a ritual designed for vampire prevention in 2004 ...)
You also ask
Is there a reason why it could not be gold or bronze?
which carries a whiff of roleplaying/worldbuilding/writers about it, but let that pass; and leaving aside just how much "[a] reason" could play into mythology:
One of the older uses of the gold hieroglyph . . . typically featured the image of a Horus falcon perched above or beside the hieroglyph for gold.
The meaning of this particular title has been disputed. One belief is that it represents the triumph of Horus over his uncle Seth, as the symbol for gold can be taken to mean that Horus was "superior to his foes". Gold also was strongly associated in the ancient Egyptian mind with eternity, so this may have been intended to convey the pharaoh's eternal Horus name.
This is easily translates to "eternal and/or divine aspect = proof against undead" ... pretty speculative though.
In some forms of Christianity and Judaism, gold has been associated both with holiness and evil. In the Book of Exodus, the Golden Calf is a symbol of idolatry, while in the Book of Genesis, Abraham was said to be rich in gold and silver, and Moses was instructed to cover the Mercy Seat of the Ark of the Covenant with pure gold. In Byzantine iconography the halos of Christ, Mary and the Christian saints are often golden.
Perhaps this ambivalence accounts for gold not being considered exclusively holy or unholy - associations with idolatry or greed vs. use with relics. But in the tales where a crucifix or other holy symbol may be used to ward off a vampire, there doesn't seem to be anything wrong with that device being golden.
Metallic and gold compounds have long been used for medicinal purposes. Gold, usually as the metal, is perhaps the most anciently administered medicine (apparently by shamanic practitioners) and known to Dioscorides. In medieval times, gold was often seen as beneficial for the health, in the belief that something so rare and beautiful could not be anything but healthy. Even some modern esotericists and forms of alternative medicine assign metallic gold a healing power.
In the 19th century gold had a reputation as a "nervine", a therapy for nervous disorders. Depression, epilepsy, migraine, and glandular problems such as amenorrhea and impotence were treated, and most notably alcoholism (Keeley, 1897).
As to 'why not bronze', well ... the mystical properties of bronze just wasn't ever 'a thing', as far as I can tell. All I see so far is references to copper associated with Venus and tin associated with Jupiter, but no substantive suggestions regarding the alloy.
So ... nothing much so far; looking forward to seeing what everyone else digs up!